For \(x ∈ R\), let \(y(x)\) be the solution of the differential equation \((x^2 -5)\)\(\frac{dy}{dx}\)\(-2xy = - 2x\)\((x^2-5)^2\) such that \(y(2) =7\). Then the maximum value of the function \(y(x)\) is
Solution and Explanation
Given:
\((x^2 -5)\)\(\frac{dy}{dx}\)-2xy = - 2x\((x^2-5)^2\)
Rearrange the equation to solve for \(\frac{dy}{dx}\):
\(\frac{dy}{dx}=\frac{2x(x^2-5)^2-2xy}{x^2-5}\)
Now, we have a first-order separable differential equation. Let's separate variables:
\(\frac{dy}{{2x(x^2-5)^2-2xy}}=\frac{dx}{x^2-5}\)
Next, integrate both sides:
\(\int\frac{dy}{{2x(x^2-5)^2-2xy}}=\int\frac{dx}{x^2-5}\)
Integrating these expressions will provide us with the function y(x).
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations